We constrain the properties of the progenitor system of the highly reddened Type Ia supernova (SN Ia) 2014J in Messier 82 (M82; d (is) approx. 3.5 Mpc). We determine the supernova (SN) location using Keck-II K-band adaptive optics images, and we find no evidence for flux from a progenitor system in pre-explosion near-ultraviolet through near-infrared Hubble Space Telescope (HST) images. Our upper limits exclude systems having a bright red giant companion, including symbiotic novae with luminosities comparable to that of RS Ophiuchi. While the flux constraints are also inconsistent with predictions for comparatively cool He-donor systems (T (is) approximately 35,000 K), we cannot preclude a system similar to V445 Puppis. The progenitor constraints are robust across a wide range of RV and AV values, but significantly greater values than those inferred from the SN light curve and spectrum would yield proportionally brighter luminosity limits. The comparatively faint flux expected from a binary progenitor system consisting of white dwarf stars would not have been detected in the pre-explosion HST imaging. Infrared HST exposures yield more stringent constraints on the luminosities of very cool (T (is) less than 3000 K) companion stars than was possible in the case of SN Ia 2011fe.